Grades
Standard
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation [...]
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For [...]
(+) Represent a system of linear equations as a single matrix equation in a vector variable.
(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using [...]
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the [...]
(+) Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that [...]
(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role [...]
(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another [...]
(+) Work with 2 X 2 matrices as transformations of the plane, and interpret the absolute value of the determinant [...]
(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are [...]
(+) Add, subtract, and multiply matrices of appropriate dimensions.
(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies [...]
Interpret expressions that represent a quantity in terms of its context.*
Interpret parts of an expression, such as terms, factors, and coefficients.*
Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)^n as [...]
Use the structure of an expression to identify ways to rewrite it. For example, see x^4 – y^4 as (x^2)^2 [...]
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the [...]
Factor a quadratic expression to reveal the zeros of the function it defines.
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
Use the properties of an expression to identify ways to rewrite it. For example, see x4 - y4 as (x2)2 [...]
Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use [...]
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, [...]
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x [...]
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the [...]
Define appropriate quantities for the purpose of descriptive modeling.*
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.*
Know there is a complex number i such that i^2 = −1, and every complex number has the form a [...]
Use the relation i^2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain [...]
(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation [...]
(+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of [...]
Solve quadratic equations with real coefficients that have complex solutions.
(+) Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x – 2i).
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate [...]
(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a [...]
(+) Solve problems involving velocity and other quantities that can be represented by vectors.
(+) Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors [...]
(+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
(+) Understand vector subtraction v – w as v + (–w), where (–w) is the additive inverse of w, with [...]
(+) Perform operations on vectors. Multiply a vector by a scalar.
(+) Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v(sub [...]
Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of [...]
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting [...]
Grades
Standard
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation [...]
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For [...]
(+) Represent a system of linear equations as a single matrix equation in a vector variable.
(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using [...]
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the [...]
(+) Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that [...]
(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role [...]
(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another [...]
(+) Work with 2 X 2 matrices as transformations of the plane, and interpret the absolute value of the determinant [...]
(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are [...]
(+) Add, subtract, and multiply matrices of appropriate dimensions.
(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies [...]
Interpret expressions that represent a quantity in terms of its context.*
Interpret parts of an expression, such as terms, factors, and coefficients.*
Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)^n as [...]
Use the structure of an expression to identify ways to rewrite it. For example, see x^4 – y^4 as (x^2)^2 [...]
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the [...]
Factor a quadratic expression to reveal the zeros of the function it defines.
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
Use the properties of an expression to identify ways to rewrite it. For example, see x4 - y4 as (x2)2 [...]
Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use [...]
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, [...]
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x [...]
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the [...]
Define appropriate quantities for the purpose of descriptive modeling.*
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.*
Know there is a complex number i such that i^2 = −1, and every complex number has the form a [...]
Use the relation i^2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain [...]
(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation [...]
(+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of [...]
Solve quadratic equations with real coefficients that have complex solutions.
(+) Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x – 2i).
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate [...]
(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a [...]
(+) Solve problems involving velocity and other quantities that can be represented by vectors.
(+) Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors [...]
(+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
(+) Understand vector subtraction v – w as v + (–w), where (–w) is the additive inverse of w, with [...]
(+) Perform operations on vectors. Multiply a vector by a scalar.
(+) Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v(sub [...]
Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of [...]
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting [...]
